Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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Norm of Fredholm operator in $L^1$

Let $T:L^1([0,1])\rightarrow L^1([0,1])$ be the Fredholm integral operator given by $$ Tf(x)=\int_0^1 k(x,y)f(y)\, dy $$ where $k \in C([0,1]^2)$ is called the kernel of $T$. My problem is to find ...
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51 views

Is everything an operator?

For example, I have some number $\alpha$ and a function $f$. Now I multiple this constant $\alpha$ with $f$ and get $\alpha * f$. Now I claim that $\alpha$ is an operator, $f$ my eigenvector, with ...
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32 views

If the scalar product are equal then the operators are equal.

I want to show the following: Let H be a $\mathbb C$ -hilbert space and $S,T\in L(X)$ If $\langle Sx,x \rangle = \langle Tx,x \rangle$ for all $x\in H$, then $S=T$ Any hints for me?
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1answer
39 views

Question about the notation $S \subset T$ ,where $S$ and $T$ are operators

I want to prove that if $S\subset T$. Then $T^{*}\subset S^{*}$. But what does $S\subset T$ mean? $S$ and $T$ are operators and not sets.. :/
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24 views

Sums of two closed and closed / continuous operators

Let $X$ be a normed space and $A_j:D(A_j)\rightarrow X$ (j=1,2) linear. (i) If $D(A_1)=X$, $A_1$ continuous and A_2 closed. Do we have $A_1+A_2:D(A_1)\cap D(A_2) \rightarrow X$, $x\mapsto A_1x+A_2x$ ...
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1answer
58 views

Unitary Equivalent of Derivative in Fourier Space

It is known that for $L^2(\mathbb R)$ the operator $Tf(x) = if'(x)$ is unitary equivalent to $\hat T \hat f(\xi )= \xi \hat f(\xi) $. Where domain of T is $H^1(\mathbb R)$. Hence the Spectrum of T in ...
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1answer
38 views

Do spectrum and Eigenvalues of $Af=-f''$ concide (under dirichlet boundary conditions)

I am asked to show that for the operator $$ Af = -f'' $$ with $D(A)=\left\{f\in H^2(0,1), f(1)=f(0)=0 \right\} \subset L^2(0,1)$ is self Adjoint in $L^2(0,1)$ (This part is solved). I cannot see ...
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35 views

For positive operators $A$ and $B$ with $A^6=B^6$ show that $A=B$

Since $A$ and $B$ are positive, I managed to show that $A^6$ and $B^6$ are positive. Now, I can use the fact that there exists a unique square root of both of those and since they're equal, their ...
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61 views

Show that the given family of bounded operators on a hilbert space form a semi group.

Suppose $A:D(A)\subset H\rightarrow H$ is a self adjoint, densly defined closed operator and it is also positive operator i.e $<Au,u>\geq0 $, for all $u\in H$ ,where $H$ is a hilbert space . ...
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35 views

Linear compact operators

Let $X$ be an infinite-dimensional Banach space, $Y$ be a Banach space, $A: X \to Y$ be a linear compact operator. Is it true that there is always a sequence $\{x_n\}\subset X$ such that $\|x_n\| \to ...
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95 views

Maximal ideal space

Let $X$ be a compact space, $x_0\in X$, and define $$A=\{\{f_n\} ; f_n\in C(X), \sup_n\|f_n\|<\infty, and \{f_n(x_0)\} \text{ is a convergent sequence} \} $$ If $\|\{f_n\}\|$ is defined as ...
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2answers
54 views

Existence fixed point

Let $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $g:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and compact valued. Consider the function $F: \mathbb{R}^n \times ...
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49 views

Invertible operators on a separable Hilbert space

Using polar decomposition or Kuiper's theorem one can show that the set of invertible operators on a separable Hilbert space $H$ is a connected subset of ${\mathcal B}(H)$. But does anyone know an ...
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40 views

A question about adjoint matrices

Let $T:V \to V $ be a linear map on complex vector space $V$ which is equipped with complex inner product $ <. , .> $ we know there exists a unique linear operator $T^* : V \to V $ such that ...
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3answers
100 views

Adjoint operator of $L^\infty$

Lets denote with $(\Omega,\Sigma,\mu)$ a $\sigma$-finite measurble space with a linear, continuous operator $$T : L^\infty \to L^\infty.$$ Does this always imply the existence of a linear, continuous ...
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34 views

Symmetries on Hilbert spaces

Let $\mathfrak{H}$ be a Hilbert space and let $\mathcal{E}(\mathfrak{H})$ be the set of all operators $T\in B(\mathfrak{H})$ such that $0\leq T\leq 1$ (these operators are also called effects on ...
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126 views

Essential self-adjointness of the Laplace operator via the Fourier transform

I'm working through some notes on showing the essential self-adjointness of the Laplace operator on $\Bbb R$ via the Fourier transform (see here) but there seems to be a little bit of liberty taken at ...
2
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1answer
51 views

Riemann manifold with unbounded Laplacian

How can one characterize a Riemann manifold the Laplacian of which is unbounded? (Equivalently, what are those manifolds on which the Laplacian is bounded? I am interested in working with its ...
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1answer
255 views

Is the injectivity of the operator equivalent to the surjectivity of its adjoint

Let $X$ and $Y$ be two normed linear spaces. Let $T:X \to Y^*$ be a linear operator (not necessarily continuous) and let $T^*$ be its adjoint, i.e. $T^*:Y \to X^*$ is defined by $ \langle T^*y,x ...
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1answer
76 views

Compact space X is totally disconnected if and only if C(X) is generated by its projections

If $X$ is compact, show that $X$ is totally disconnected if and only if $C(X)$ as a C*-algebra is generated by its projections. My attempt: Suppose $X$ is totally disconnected, then $X=\{x_i\}_{i\in ...
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1answer
79 views

Linear operator norm

I am trying to show that these two definitions for a bounded linear operator norm on the normed linear space $X$ are equivalent: $$ \sup\{T(x)\,:\, \|x\|\le 1\}=\|T\|_*=\inf\{M>0\,:\, T(x)\le ...
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1answer
36 views

A non-continuous idempotent linear operator in a Banach space

Does there exist a non-continuous idempotent linear operator $P: X \to X$ where $X$ is an infinite-dimensional Banach space? That is, $P^2 = P$, and there is a sequence $\{x_n\}$ of elements of X such ...
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52 views

Orthogonal projectors on non-orthogonal subspaces

It is a well known fact that if(f) $V,W$ are orthogonal subspaces of a Hilbert space $H$, then their orthogonal projectors satisfy: $$ P_{VW} = P_V + P_W, $$ where $P_{VW}$ is the projector on $V+W$. ...
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77 views

Strong resolvent convergence and spectral measures

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in the strong resolvent sense. Denoting by $E_n$ and $E$ ...
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57 views

Discrete bi-Laplacian

I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges. In particular ...
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18 views

References on projectors

What are good books or articles about linear projectors in Hilbert spaces? I am mostly interested in the finite dimensional case (but anything is welcome). All about idempotents, orthogonal and ...
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78 views

Perturbation of Laplacian

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; ...
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1answer
42 views

Proving equivalence of two definitions

Hi all I am intersted in proving the equivalence of the following two definitions of pseudomontoncity: Let $V$ be a reflexive Banach space and $K \subset V$ closed and convex. Definition 1: $A: V ...
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1answer
31 views

the sum of two unbounded normal operators

why A and B are normal?and why "0" is not closed on H1(R)?
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50 views

Approximation Property: Decomposition

This is a real question of me. Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional range: ...
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1answer
27 views

Question about linear operator continuity

Let $A:X\rightarrow Y$ be a linear operator, $X,Y$ normed spaces. Show that a linear operator is continuous (bonded) if for every sequence $x_n\rightarrow 0$ in $X$ has a bounded image $Ax_n$ in $Y$. ...
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1answer
99 views

Does this operator equation have solutions?

Hi Math StackExchange community, I have a question that originates from a Physics problem; the question itself however is about solving an operator equation. In a particular quantum mechanical ...
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1answer
50 views

How do I prove this statement about the operator norm?

I stumbled across this equation in a paper, which may seem obvious, but I'm wondering if someone can explain why this is true? By definition of an operator norm, $$\left[(D^*D)^{-1} - ...
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1answer
64 views

Approximation Property: Characterization

As reference the german wiki: Approximationseigenschaft Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$ ...
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1answer
50 views

Inverse operator of $I-A$

Let $H$ be an Hilbert space, $A:H\to H$ be a bounded linear operator such that $$ \|A^{n_0}\|< 1\qquad\text{for some}\quad\; n_0\in\mathbb{N}. $$ I have to show that $I-A$ is invertible. My idea ...
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74 views

Partial Isometries: Subspaces

This thread was only Q&A. Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$J\in\mathcal{B}(\mathcal{H},\mathcal{K}):\quad P:=J^*J$$ By a previous thread:* ...
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1answer
45 views

Adjoints of operators between different Hilbert spaces.

When we have an operator $$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$ from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint ...
2
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1answer
38 views

Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to ...
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1answer
44 views

Separating and cyclic vector

Let $\{\Gamma_i , \mu_i\}_{i\in I}$ be a family of probability measure spaces and suppose $I$ is uncountable. Let $\{\Gamma , \mu\} = \prod_{i\in I} \{\Gamma_i,\mu_i\}$ be the product measure space. ...
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1answer
34 views

Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank ...
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2answers
65 views

Is this a bounded linear map?

I tried very hard to (dis)prove it, but now I give up. Define a map which maps $x\in L_2[0,1]$ to the function $$(Tx)(t) = \frac{1}{\sqrt{t}}\int_0^t \frac{x(s)}{\sqrt{s}} \,d s.$$ I don't even ...
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Representation of linear functional on $c$

On the space of convergent sequences $c$ let $x=(x_i)_{i\in \Bbb N}\in c$ and $\lim_{i \to \infty}x_i=x_0$ then a bonded linear functional on $c$ has a representation ...
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1answer
21 views

Action of projections

Suppose we have a projection $p$ on a Hilbert space $\mathfrak{H}$. Is the following true: There exists an set $V\subset\mathfrak{H}$ such that $p(x)=x$ if $x\in V$ and zero else? I asked because I ...
3
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1answer
70 views

Orthogonality of projections on a Hilbert space

Assume that $p$ and $q$ are (orthogonal) projections on Hilbert space $\mathcal{H}$. I want to prove: $pq=0$ iff $p+q\leq1$ I had the following in mind: Assume $pq=0$. Then $qp=0$, hence $p+q$ is a ...
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1answer
21 views

Image of bounded linear operator?

Let $x^\ast$ be a continuous linear functionals on $l_p$. Let $(e_i)_{i\in \Bbb N}$ be the standard basis of $l_p$. Consider $y=(y_i)_{i\in \Bbb N}$ the sequence defined by $y_i=x^\ast(e_i)$. Let ...
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2answers
56 views

Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
2
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1answer
35 views

Is this a bounded linear operator?

I have the following problem. Show that $y_n={1 \over \sqrt n}\int_0^1t^nx(t)dt$ is a bounded linear operator that maps $L_2[0,1]$ into $l_2$ with the usual norm on the respective spaces. My approach ...
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1answer
140 views

spectrum of convolution integral operator

Let $A f(x)= \int_{-\pi}^{\pi} h(x-y) f(y) dy$ operator $L^2( {-\pi},{\pi})->L^2( {-\pi},{\pi}), h$ is continuous, periodic with period $2\pi$ and $h(x)=h(-x)$ on $ [ {-\pi},{\pi}] $. How can I ...
2
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2answers
75 views

Self-adjoint operator- domain unique?

I was wondering about the following: Let $T : dom(T) \subset H \rightarrow H$ be a self-adjoint operator, does this mean that the domain of $T$ is uniquely defined or is it possible to make the same ...
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1answer
32 views

characterizing an operator with projection whose spectrum is contained in $\{-1,1\}$

Let $\mathcal{A}$ be a $C^{*}$-algebra and $\sigma$ denote the spectrum. I want to show that if $\sigma (A)\subseteq \{-1,+1\}$ for $A\in \mathcal{A}$ then there is a projection $P$ such that ...